Abstract
The Gehring–Martin–Tan number and the Tan number are real quantities defined for two-generated subgroups of the group PSL(2,ℂ). It follows from the necessary discreteness conditions proved by Gehring and Martin and, independently, by Tan that, for discrete groups, these quantities are bounded below by 1. In the paper, we find precise values of these numbers for the majority of elementary discrete groups and prove that, for every real r ≥ 1, there are infinitely many elementary discrete groups with the Gehring–Martin–Tan number equal to r and the Tan number equal to r.
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References
T. Jørgensen, “A note on subgroups of SL(2, C),” Quart. J. Math. Oxford Ser. (2) 28 (110), 209–211 (1977).
A. F. Beardon, The Geometry of Discrete Groups (Academic Press, London, 1977; Nauka, Moscow, 1986).
T. Jørgensen, “On discrete groups of Möbius transformations,” Amer. J. Math. 98 (3), 739–749 (1976).
A. Yu. Vesnin and A. V. Maslei, Two-Generated Subgroups of PSL(2, C) Extremal for the Jørgensen Inequality and Its Analogs, in Proceedings of the Seminar on Vector and Tensor Analysis and Its Applications to Geometry, Mechanics, and Physics (Izd. Moskov. Univ., Moscow, 2014), Vol. 30 [in Russian].
T. Jørgensen and M. Kiikka, “Some extreme discrete groups,” Ann. Acad. Sci. Fenn. Ser. A I, Math. 1 (2), 245–248 (1975).
T. Jørgensen, A. Lascurain, and T. Pignataro, “Translation extensions of the classical modular group,” Complex Variables Theory Appl. 19 (4), 205–209 (1992).
H. Sato and R. Yamada, “Some extreme Kleinian groups for Jorgensen’s inequality,” Rep. Fac. Sci. Shizuoka Univ. 27, 1–8 (1993).
H. Sato, “One-parameter families of extreme discrete groups for Jørgensen’s inequality,” in In the Tradition of Ahlfors and Bers, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2000), Vol. 256, pp. 271–287.
B. Maskit, “Some special 2-generator Kleinian groups,” Proc. Amer. Math. Soc. 106 (1), 175–186 (1989).
R. González-Acuna and A. Ramírez, “Jørgensen subgroups of the Picard group,” Osaka J. Math. 44 (2), 471–482 (2007).
J. Callahan, “Jørgensen number and arithmeticity,” Conform. Geom. Dyn. 13, 160–186 (2009).
F. W. Gehring and G. J. Martin, “Stability and extremality in Jørgensen’s inequality,” Complex Variables Theory Appl. 12 (1-4), 277–282 (1989).
C. Li, M. Oichi, and H. Sato, “Jørgensen groups of parabolic type I (finite case),” Comput. Methods Funct. Theory. 5 (2), 409–430 (2005).
C. Li, M. Oichi, and H. Sato, “Jørgensen groups of parabolic type II (countably infinite case),” Osaka J. Math. 41 (3), 491–506 (2004).
C. Li, M. Oichi, and H. Sato, “Jørgensen groups of parabolic type III (uncountably infinite case),” Kodai Math. J. 28 (2), 248–264 (2005).
H. Sato, “The Jørgensen number of the Whitehead link group,” Bol. Soc. Mat. Mexicana (3) 10, 495–502 (2004).
M. Oichi and H. Sato, “Jørgensen numbers of discrete groups,” Complex Analysis and Geometry of Hyperbolic Spaces 1518, 105–118 (2006).
A. Yu. Vesnin and A. V. Masley [A. V. Maslei], “On Jørgensen numbers and their analogs for groups of figure-eight orbifolds,” Sibirsk. Mat. Zh. 55 (5), 989–1000 (2014) [Sib. Math. J. 55 (5),), 807–816 (2014].
F. W. Gehring and G. J. Martin, “Iteration theory and inequalities for Kleinian groups,” Bull. Amer. Math. Soc. (N. S.) 21 (1), 57–63 (1989).
D. Tan, “On two-generator discrete groups of Möbius transformations,” Proc. Amer. Math. Soc. 106 (3), 763–770 (1989).
J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, in Grad. Texts in Math. (Springer-Verlag, New York, 2006), Vol. 149.
D. Repovš and A. Vesnin, “On Gehring–Martin–Tan groups with an elliptic generator,” Bull. Aust. Math. Soc. 94 (2), 326–336 (2016).
N. A. Isachenko, “Systems of generators of subgroups of PSL(2, C),” Sibirsk. Mat. Zh. 31 (1), 191–193 (1990) [Sib. Math. J. 31 (1), 162–165 (1990)].
A. V. Maslei, “On parameters and discreteness of Maskit subgroups in PSL(2, C),” Sib. Elektron. Matem. Izv. 14, 125–134 (2017).
É. B. Vinberg, A Course in Algebra (Faktorial Press, Moscow, 2001; American Mathematical Society, Providence, RI, 2003).
F. W. Gehring and G. J. Martin, “Commutators, collars and the geometry of Möbius groups,” J. Anal. Math. 63 (1), 175–219 (1994).
H. S. M. Coxeter, Regular Polytopes (Methuen, London, 1948).
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Original Russian Text © A. V. Maslei, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 2, pp. 255–269.
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Maslei, A.V. Gehring–Martin–Tan numbers and Tan numbers of elementary subgroups of PSL(2,ℂ). Math Notes 102, 219–231 (2017). https://doi.org/10.1134/S0001434617070240
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DOI: https://doi.org/10.1134/S0001434617070240